No effect on Idol Toss. Idol Toss chance is entirely proportional to how many idols you have and how many eggs you use.

No better loot. Weapon / power quality depends on intensity and what you have unlocked in the treasure hunt.

Fireworks don't do more damage.

Additionally, for those who have beaten the game:

Later in the game, the UI switches to Viridi instead of Palutena. After you beat the game, you can use the Hidden Options to switch between the two. They have different donation pools. Neither does anything.

Okay, I have finally gotten one million hearts, and i will be giving it all ot palutena in 50 thousand increments.

So after giving her 1 million hearts... she is pretty close to the screen. I can only see from the waist up. I checked the prices of the weapons that I can buy and sell, and there is no notable difference. Fusing weapons doesn't add anything new, either.

What I did find, however, is that when I played a chapter, I found a bit of new dialogue between Pit and Palutena. I was playing chapter three, and Pit randomly began to talk about his goldfish. This may or may not be related to the heart offering or the intensity of the level(I was playing it at 8.5).

Also, the amount of hearts I could get in a level did not change. However, when I obtained weapons from chests, they seemed to be better than they usually were, such as more frequent weapons in the 300 value range with good ranged, melee, and attributes. That is about all I can see at the moment. Hope this cleared things up a bit.

The new dialogue is related to intensity, and how much time the level is taking. I've found it playing at both lower and higher intensities (and I think I've heard the goldfish one, and I'm nowhere near 1,000,000 hearts). The value of items in chests is tied directly to intensity. I find ~200-250 value loot playing at intensity 7, so 300 at 8.5 doesn't seem too far out of whack - I don't think it's goddess-related. You should've compared before and after loot value to be sure, but I guess it's too late.
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user2640Apr 2 '12 at 16:18

My mistake on my previous answer. I guess it was just coincidence of when I bought weapons. However, having now finally donated one million+ hearts, I think I found what it REALLY does. Apparently, assuming my theory is correct, as you donate more hearts, better weapons can be found. For example, donating hearts can unlock effects on weapons, such as Petrification, or at least increase their likelihood. Plus, it seems to have an effect on the weapon's overall value, as well. For example, before I donated the million, I could only find weapons at 200 tops or so. But now, I can find weapons at 300 and above. Hope this helps!

More dialogue appear between Pit and Palutena!
I found that out today(as of April 15, 2012) on Ch.6 or something(where you fight Dark Pit on the floating platforms) and when Pit is going up the rail, Pit and Palutena sing together which hasn't happened before offering hearts. It was really funny. And better value in weapons appear. I offered like 93,000 hearts and im definitely seeing something different every time I play on Solo. Hope this help you guys out.
-Oscar

Donating hearts give you a better "SALE" on the item.
i went in to the store and saw an wolf claw for 75k
i didnt buy it yet so i donated some hearts (i started with 0 hearts. I donated 50k hearts and my total hearts donated is 50k.)
When i went back to the store, the claws were now 30k.

Can you post some of the data that led you to this conclusion?
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user2640Mar 24 '12 at 16:33

-1 I've done some testing and the shop prices don't change if I offer her hearts. What lead you to this conclusion?
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WipqoznMar 25 '12 at 20:40

I've seen this statement made by a couple different users on other forums as well. Maybe it just needs a ridiculous number of hearts, or follows some super-linear curve? Shame I've already donated 30k and not paid attention before that...
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user2640Mar 25 '12 at 20:57

I went from 30k to 70k with no notable effect, so I've got to downvote this.
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user2640Mar 29 '12 at 19:57